Choice A is correct. It's given that a rectangle has a length that is $15$ times its width. It's also given that the function $y=\left(15w\right)\left(w\right)$ represents this situation, where $y$ is the area, in square feet, of the rectangle and $y>0$. The area of a rectangle can be calculated by multiplying the rectangle's length by its width. Since the rectangle has a length that is $15$ times its width, it follows that $w$ represents the width of the rectangle, in feet, and $15w$ represents the length of the rectangle, in feet. Therefore, the best interpretation of $15w$ in this context is that it's the length of the rectangle, in feet.

Choice B is incorrect. This is the best interpretation of $y$, not $15w$, in the given function.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect. This is the best interpretation of $w$, not $15w$, in the given function.

Choice A is correct. It’s given that $x$ is positive. Therefore, multiplying each side of the given equation by $-150x$ yields $-150xv=w$, which is equivalent to $w=-150vx$. Thus, the equation $w=-150vx$ correctly expresses $w$ in terms of $v$ and $x$.

Choice B is incorrect. This equation is equivalent to $v=-\frac{wx}{150}$.

Choice C is incorrect. This equation is equivalent to $v=-\frac{x}{150w}$.

Choice D is incorrect. This equation is equivalent to $v=w-150x$.

Choice B is correct. Applying the commutative property of multiplication, the expression $\left(m^4{q}^4{z}^{-1}\right)\left(m{q}^5{z}^3\right)$ can be rewritten as $\left(m^{4}m\right)\left(q^4{q}^5\right)\left(z^{-1}{z}^3\right)$. For positive values of $x$, $\left(x^a\right)\left(x^b\right)=x^{a+b}$. Therefore, the expression $\left(m^{4}m\right)\left(q^4{q}^5\right)\left(z^{-1}{z}^3\right)$ can be rewritten as $\left(m^{4+1}\right)\left(q^{4+5}\right)\left(z^{-1+3}\right)$, or $m^5{q}^9{z}^2$.

Choice A is incorrect and may result from multiplying, not adding, the exponents.

Choice C is incorrect and may result from conceptual or calculation errors. 

Choice D is incorrect and may result from conceptual or calculation errors. 

Choice C is correct. The solutions to a system of two equations correspond to points where the graphs of the equations intersect. The given graphs intersect at 2 points; therefore, the system has 2 solutions.

Choice A is incorrect because the graphs intersect. Choice B is incorrect because the graphs intersect more than once. Choice D is incorrect. It’s not possible for the graph of a quadratic equation and the graph of a linear equation to intersect more than twice.

Choice B is correct. Both sides of the given inequality can be divided by 3 to yield .

Choices A, C, and D are incorrect because they are not equivalent to (do not have the same solution set as) the given inequality. For example, the ordered pair is a solution to the given inequality, but it is not a solution to any of the inequalities in choices A, C, or D.

Choice A is correct. It is given that y = x + 1 and y = x² + x. Setting the values for y equal to each other yields x + 1 = x² + x. Subtracting x from each side of this equation yields x² = 1. Therefore, x can equal 1 or –1. Of these, only –1 is given as a choice.

Choice B is incorrect. If x = 0, then x + 1 = 1, but x² + x = 0² + 0 = 0 ≠︀ 1. Choice C is incorrect. If x = 2, then x + 1 = 3, but x² + x = 2² + 2 = 6 ≠︀ 3. Choice D is incorrect. If x = 3, then x + 1 = 4, but x² + x = 3² + 3 = 12 ≠︀ 4.

Choice B is correct. Since $2xy$ is a common factor of each term in the given expression, the expression can be rewritten as $2xy\left(8x^{2}y+7\right)$.

Choice A is incorrect. This expression is equivalent to $16x^2{y}^2+14xy$.

Choice C is incorrect. This expression is equivalent to $28x^3{y}^2+14xy$.

Choice D is incorrect. This expression is equivalent to $112x^3{y}^2+14xy$.

Choice B is correct. The first and last terms of the polynomial are both squares such that  and . The second term is twice the product of the square root of the first and last terms: . Therefore, the polynomial is the square of a binomial such that , and  is a factor.

Choice A is incorrect and may be the result of incorrectly factoring the polynomial. Choice C is incorrect and may be the result of dividing the second and third terms of the polynomial by their greatest common factor. Choice D is incorrect and may be the result of not factoring the coefficients.

Choice B is correct. It’s given that and . Substituting the values for p and v into the expression yields . Multiplying the terms yields . Using the distributive property to rewrite yields . Therefore, the entire expression can be represented as . Combining like terms yields .

Choice A is incorrect and may result from subtracting, instead of adding, the term . Choice C is incorrect. This is the result of multiplying the terms . Choice D is incorrect and may result from distributing 2, instead of , to the term .

Choice C is correct. Using the quadratic formula to solve the given expression yields . Therefore, and satisfy the given equation.

Choices A and B are incorrect and may result from incorrectly factoring or incorrectly applying the quadratic formula. Choice D is incorrect and may result from a sign error.

Choice B is correct. The number of solutions to a quadratic equation of the form $a{x}^2+bx+c=0$, where $a$, $b$, and $c$ are constants, can be determined by the value of the discriminant, $b^2-4ac$. If the value of the discriminant is positive, then the quadratic equation has exactly two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is negative, then the quadratic equation has zero real solutions. In the given equation, $3x^2-15x+18=0$, $a=3$, $b=-15$, and $c=18$. Substituting $3$ for $a$, $-15$ for $b$, and $18$ for $c$ in $b^2-4ac$ yields $(-15{)}^2-4(3\left)\right(18)$, or $9$. Since the value of the discriminant is positive, the given equation has exactly two distinct real solutions.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is $14$. Let $x$ represent the first integer and $y$ represent the second integer. If the first integer is $5$ greater than twice the second integer, then $x=2y+5$. It's given that the product of the two integers is $462$; therefore $xy=462$. Substituting $2y+5$ for $x$ in this equation yields $(2y+5)\left(y\right)=462$, which can be written as $2y^2+5y=462$. Subtracting $462$ from each side of this equation yields $2y^2+5y-462=0$. The left-hand side of this equation can be factored by finding two values whose product is $2(-462)$, or $-924$, and whose sum is $5$. The two values whose product is $-924$ and whose sum is $5$ are $33$ and $-28$. Thus, the equation $2y^2+5y-462=0$ can be rewritten as $2y^2-28y+33y-462=0$, which is equivalent to $2y(y-14)+33(y-14)=0$, or $(2y+33)(y-14)=0$. By the zero product property, it follows that $2y+33=0$ or $y-14=0$. Subtracting $33$ from both sides of the equation $2y+33=0$ yields $2y=-33$. Dividing both sides of this equation by $2$ yields $y=-\frac{33}{2}$. Since $y$ is a positive integer, the value of $y$ isn't $-\frac{33}{2}$. Adding $14$ to both sides of the equation $y-14=0$ yields $y=14$. Substituting $14$ for $y$ in the equation $xy=462$ yields $x\left(14\right)=462$. Dividing both sides of this equation by $14$ yields $x=33$. Therefore, the two integers are $14$ and $33$, so the smaller of the two integers is $14$.
 

The correct answer is $609$. It’s given that the function $g$ is defined by $g\left(x\right)=\frac{\left|x\right|}{a}-14$, where $a<0$. Substituting $15a$ for $x$ in function $g$ yields $g\left(15a\right)=\frac{\left|15a\right|}{a}-14$. This function can be rewritten as $g\left(15a\right)=\frac{15\left|a\right|}{a}-14$, or $g\left(15a\right)=15\left(\frac{\left|a\right|}{a}\right)-14$. Since $a<0$, it follows that $\frac{\left|a\right|}{a}=-1$. Substituting $-1$ for $\frac{\left|a\right|}{a}$ in $g\left(15a\right)=15\left(\frac{\left|a\right|}{a}\right)-14$ yields $g\left(15a\right)=15(-1)-14$, or $g\left(15a\right)=-29$. Similarly, substituting $7a$ for $x$ in function $g$ yields $g\left(7a\right)=\frac{\left|7a\right|}{a}-14$. This function can be rewritten as $g\left(7a\right)=\frac{7\left|a\right|}{a}-14$, or $g\left(7a\right)=7\left(\frac{\left|a\right|}{a}\right)-14$. Since $a<0$, it again follows that $\frac{\left|a\right|}{a}=-1$. Substituting $-1$ for $\frac{\left|a\right|}{a}$ in $g\left(7a\right)=7\left(\frac{\left|a\right|}{a}\right)-14$ yields $g\left(7a\right)=7(-1)-14$, or $g\left(7a\right)=-21$. Therefore, $g\left(15a\right)=-29$ and $g\left(7a\right)=-21$. Thus, the product of $g\left(15a\right)$ and $g\left(7a\right)$ is $(-29)(-21)$, or $609$.

Choice A is correct. It's given that the graph shown gives the estimated value $y$, in dollars, of a tablet as a function of the number of months since it was purchased, $x$. The y-intercept of a graph is the point at which the graph intersects the y-axis, or when $x$ is $0$. The graph shown intersects the y-axis at the point $\left(0,225\right)$. It follows that $0$ months after the tablet was purchased, or when the tablet was purchased, the estimated value of the tablet was $225$ dollars. Therefore, the best interpretation of the y-intercept is that the estimated value of the tablet was $\$225$ when it was purchased.

Choice B is incorrect. The estimated value of the tablet $24$ months after it was purchased was $\$50$, not $\$225$.

Choice C is incorrect. The estimated value of the tablet had decreased by $\$225-\$50$, or $\$175$, not $\$225$, in the $24$ months after it was purchased.

Choice D is incorrect and may result from conceptual errors.

Choice B is correct. Using the associative and commutative properties of addition, the given expression can be rewritten as . Adding these like terms results in .

Choice A is incorrect and may result from adding . Choice C is incorrect and may result from adding . Choice D is incorrect and may result from adding .

Choice A is correct. It's given that the table shows values of $x$ and their corresponding values of $g\left(x\right)$, where $g\left(x\right)=\frac{f\left(x\right)}{x+3}$. It's also given that $f$ is a linear function. It follows that an equation that defines $f$ can be written in the form $f\left(x\right)=mx+b$, where $m$ represents the slope and $b$ represents the y-coordinate of the y-intercept $\left(0,b\right)$ of the graph of $y=f\left(x\right)$ in the xy-plane. The slope of the graph of $y=f\left(x\right)$ can be found using two points, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, that are on the graph of $y=f\left(x\right)$, and the formula $m=\frac{y_2-y_1}{x_2-x_1}$. Since the table shows values of $x$ and their corresponding values of $g\left(x\right)$, substituting values of $x$ and $g\left(x\right)$ in the equation $g\left(x\right)=\frac{f\left(x\right)}{x+3}$ can be used to define function $f$. Using the first pair of values from the table, $x=-27$ and $g\left(x\right)=3$, yields $3=\frac{f\left(-27\right)}{-27+3}$, or $3=\frac{f\left(-27\right)}{-24}$. Multiplying each side of this equation by $-24$ yields $-72=f\left(-27\right)$, so the point $\left(-27,-72\right)$ is on the graph of $y=f\left(x\right)$. Using the second pair of values from the table, $x=-9$ and $g\left(x\right)=0$, yields $0=\frac{f\left(-9\right)}{-9+3}$, or $0=\frac{f\left(-9\right)}{-6}$. Multiplying each side of this equation by $-6$ yields $0=f\left(-9\right)$, so the point $\left(-9,0\right)$ is on the graph of $y=f\left(x\right)$. Substituting $\left(-27,-72\right)$ and $\left(-9,0\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the formula $m=\frac{y_2-y_1}{x_2-x_1}$ yields $m=\frac{0-\left(-72\right)}{-9-\left(-27\right)}$, or $m=4$. Substituting $4$ for $m$ in the equation $f\left(x\right)=mx+b$ yields $f\left(x\right)=4x+b$. Since $0=f\left(-9\right)$, substituting $-9$ for $x$ and $0$ for $f\left(x\right)$ in the equation $f\left(x\right)=4x+b$ yields $0=4\left(-9\right)+b$, or $0=-36+b$. Adding $36$ to both sides of this equation yields $36=b$. It follows that $36$ is the y-coordinate of the y-intercept $\left(0,b\right)$ of the graph of $y=f\left(x\right)$. Therefore, the y-intercept of the graph of $y=f\left(x\right)$ is $\left(0,36\right)$.

Choice B is incorrect. $12$ is the y-coordinate of the y-intercept of the graph of $y=g\left(x\right)$.

Choice C is incorrect. $4$ is the slope of the graph of $y=f\left(x\right)$.

Choice D is incorrect. $-9$ is the x-coordinate of the x-intercept of the graph of $y=f\left(x\right)$.

Choice B is correct. The given expression has a common factor of $2$ in the denominator, so the expression can be rewritten as $\frac{8x\left(x-7\right)-3\left(x-7\right)}{2\left(x-7\right)}$. The three terms in this expression have a common factor of $\left(x-7\right)$. Since it's given that $x>7$, $x$ can't be equal to $7$, which means $\left(x-7\right)$ can't be equal to $0$. Therefore, each term in the expression, $\frac{8x\left(x-7\right)-3\left(x-7\right)}{2\left(x-7\right)}$, can be divided by $\left(x-7\right)$, which gives $\frac{8x-3}{2}$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. It's given that the graphs of the given equations intersect at the point $\left(x,y\right)$ in the xy-plane. It follows that $\left(x,y\right)$ represents a solution to the system consisting of the given equations. The first equation given is $x=49$. Substituting $49$ for $x$ in the second equation given, $y=\sqrt{x}+9$, yields $y=\sqrt{49}+9$, which is equivalent to $y=7+9$, or $y=16$. It follows that the graphs of the given equations intersect at the point $\left(49,16\right)$. Therefore, the value of $y$ is $16$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. It’s given that the graph of intersects the x-axis at and , where t is a constant. Since this graph intersects the x-axis when or when , it follows that and . If , then . Adding 32 to both sides of this equation yields . Dividing both sides of this equation by 2 yields . Taking the square root of both sides of this equation yields . Adding 4 to both sides of this equation yields . Therefore, the value of t is 8.

Choices A, B, and C are incorrect and may result from calculation errors.

Choice A is correct. When a graph is translated up $4$ units, each point on the resulting graph is $4$ units above the point on the original graph. In other words, the y-value of each point on the graph increases by $4$. The graph shown passes through the points $\left(1,-1\right)$, $\left(2,-2\right)$, and $\left(3,-1\right)$. It follows that when the graph shown is translated up $4$ units, the resulting graph will pass through the points $\left(1,-1+4\right)$, $\left(2,-2+4\right)$, and $\left(3,-1+4\right)$. These points are $\left(1,3\right)$, $\left(2,2\right)$, and $\left(3,3\right)$, respectively. Of the given choices, only the graph in choice A passes through the points $\left(1,3\right)$, $\left(2,2\right)$, and $\left(3,3\right)$.

Choice B is incorrect. This is the result of translating the graph down, rather than up, $4$ units.

Choice C is incorrect. This is the result of translating the graph left, rather than up, $4$ units.

Choice D is incorrect. This is the result of translating the graph right, rather than up, $4$ units.

The correct answer is $\mathrm{1,728}$. The given function can be rewritten in the form $f\left(x\right)=a{\left(x-h\right)}^2+k$, where $a$ is a positive constant and the minimum value, $k$, of the function occurs when the value of $x$ is $h$. By completing the square, $f\left(x\right)=x^2-48x+\mathrm{2,304}$ can be written as $f\left(x\right)=x^2-48x+{\left(\frac{48}{2}\right)}^2+\mathrm{2,304}-{\left(\frac{48}{2}\right)}^2$, or $f\left(x\right)={\left(x-24\right)}^2+\mathrm{1,728}$. This equation is in the form $f\left(x\right)=a{\left(x-h\right)}^2+k$, where $a=1$, $h=24$, and $k=\mathrm{1,728}$. Therefore, the minimum value of the given function is $\mathrm{1,728}$.

The correct answer is $9$. The given equation can be rewritten as $5\left|4-x\right|=25$. Dividing each side of this equation by $5$ yields $\left|4-x\right|=5$. By the definition of absolute value, if $\left|4-x\right|=5$, then $4-x=5$ or $4-x=-5$. Subtracting $4$ from each side of the equation $4-x=5$ yields $-x=1$. Dividing each side of this equation by $-1$ yields $x=-1$. Similarly, subtracting $4$ from each side of the equation $4-x=-5$ yields $-x=-9$. Dividing each side of this equation by $-1$ yields $x=9$. Therefore, since the two solutions to the given equation are $-1$ and $9$, the positive solution to the given equation is $9$.

Choice A is correct. The y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane occurs at the point on the graph where $x=0$. In other words, when $x=0$, the corresponding value of $f\left(x\right)$ is the y-coordinate of the y-intercept. Substituting $0$ for $x$ in the given equation yields $f\left(0\right)=\left(-8\right){\left(2\right)}^0+22$, which is equivalent to $f\left(0\right)=\left(-8\right)\left(1\right)+22$, or $f\left(0\right)=14$. Thus, when $x=0$, the corresponding value of $f\left(x\right)$ is $14$. Therefore, the y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane is $\left(0,14\right)$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This could be the y-intercept for $f\left(x\right)=\left(-8\right){\left(2\right)}^x$, not $f\left(x\right)=\left(-8\right){\left(2\right)}^x+22$.

Choice A is correct. It's given that the graph shows the height above ground, in meters, of a ball $x$ seconds after the ball was launched upward from a platform. In the graph shown, the x-axis represents time, in seconds, and the y-axis represents the height of the ball above ground, in meters. It follows that for the marked point $\left(1.0,4.8\right)$, $1.00$ represents the time, in seconds, after the ball was launched upward from a platform and $4.80$ represents the height of the ball above ground, in meters. Therefore, the best interpretation of the marked point $\left(1.0,4.8\right)$ is $1.00$ second after being launched, the ball's height above ground is $4.80$ meters.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Choice B is correct. It’s given that the situation can be modeled by the equation $y=-100x^2+\mathrm{1,400}x+\mathrm{7,200}$, where $x$ represents the increase in ticket price, in dollars, and $y$ represents the revenue, in dollars, from ticket sales. Since the coefficient of the $x^2$ term is negative, the graph of this equation in the xy-plane opens downward and reaches its maximum value at its vertex. If a quadratic equation in the form $y=a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants, is graphed in the xy-plane, the x-coordinate of the vertex is equal to $-\frac{b}{2a}$. For the equation $y=-100x^2+\mathrm{1,400}x+\mathrm{7,200}$, $a=-100$, $b=\mathrm{1,400}$, and $c=\mathrm{7,200}$. It follows that the x-coordinate of the vertex is $-\frac{\mathrm{1,400}}{2(-100)}$, or $7$. Therefore, if the given equation is graphed in the xy-plane, the maximum of the graph occurs at an x-value of $7$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice B is correct. Adding $9$ to each side of the given equation yields $x^2-2x=9$. To complete the square, adding $1$ to each side of this equation yields $x^2-2x+1=9+1$, or ${\left(x-1\right)}^2=10$. Taking the square root of each side of this equation yields $x-1=\pm \sqrt{10}$. Adding $1$ to each side of this equation yields $x=1\pm \sqrt{10}$. Since it's given that one of the solutions to the equation can be written as $1+\sqrt{k}$, the value of $k$ must be $10$.

Alternate approach: It's given that $1+\sqrt{k}$ is a solution to the given equation. It follows that $x=1+\sqrt{k}$. Substituting $1+\sqrt{k}$ for $x$ in the given equation yields ${\left(1+\sqrt{k}\right)}^2-2\left(1+\sqrt{k}\right)-9=0$, or $\left(1+\sqrt{k}\right)\left(1+\sqrt{k}\right)-2\left(1+\sqrt{k}\right)-9=0$. Expanding the products on the left-hand side of this equation yields $1+2\sqrt{k}+k-2-2\sqrt{k}-9=0$, or $k-10=0$. Adding $10$ to each side of this equation yields $k=10$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice B is correct. For a quadratic function defined by an equation of the form $p\left(x\right)=a{\left(x-h\right)}^2+k$, where $a$, $h$, and $k$ are constants and $a>0$, the minimum value of the function is $k$. Subtracting $57$ from both sides of the given equation yields $p\left(x\right)=x^2-57$. This function is in the form $p\left(x\right)=a{\left(x-h\right)}^2+k$, where $a=1$, $h=0$, and $k=-57$. Therefore, the minimum value of the function $p$ is $-57$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is $6$. It’s given that a line with equation $2y=4.5$ intersects a parabola with equation $y=-4x^2+bx$, where $b$ is a positive constant, at exactly one point in the xy-plane. It follows that the system of equations consisting of $2y=4.5$ and $y=-4x^2+bx$ has exactly one solution. Dividing both sides of the equation of the line by $2$ yields $y=2.25$. Substituting $2.25$ for $y$ in the equation of the parabola yields $2.25=-4x^2+bx$. Adding $4x^2$ and subtracting $bx$ from both sides of this equation yields $4x^2-bx+2.25=0$. A quadratic equation in the form of $a{x}^2+bx+c=0$, where $a$, $b$, and $c$ are constants, has exactly one solution when the discriminant, $b^2-4ac$, is equal to zero. Substituting $4$ for $a$ and $2.25$ for $c$ in the expression $b^2-4ac$ and setting this expression equal to $0$ yields $b^2-4\left(4\right)(2.25)=0$, or $b^2-36=0$. Adding $36$ to each side of this equation yields $b^2=36$. Taking the square root of each side of this equation yields $b=\pm 6$. It’s given that $b$ is positive, so the value of $b$ is $6$.

Choice D is correct. The y-intercept of a graph in the xy-plane is the point where $x=0$. For the given function $f$, the y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane can be found by substituting $0$ for $x$ in the equation $y=\mathrm{9,000}{\left(0.66\right)}^x$, which gives $y=\mathrm{9,000}{\left(0.66\right)}^0$. This is equivalent to $y=\mathrm{9,000}\left(1\right)$, or $y=\mathrm{9,000}$. Therefore, the y-intercept of the graph of $y=f\left(x\right)$ is $\left(0,\mathrm{9,000}\right)$. It’s given that the function $f$ models the number of advertisements a company sent to its clients each year. Therefore, $f\left(x\right)$ represents the estimated number of advertisements the company sent to its clients each year. It's also given that $x$ represents the number of years since $1997$. Therefore, $x=0$ represents $0$ years since $1997$, or $1997$. Thus, the best interpretation of the y-intercept of the graph of $y=f\left(x\right)$ is that the estimated number of advertisements the company sent to its clients in $1997$ was $\mathrm{9,000}$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The given expression can be rewritten as $\left(9x^3+6x^3\right)+5x^2+5x+\left(7-5\right)$. Combining like terms in this expression yields $15x^3+5x^2+5x+2$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is correct. Since $w$ represents the $n$th term of the sequence and $9$ is the first term of the sequence, the value of $w$ is $9$ when the value of $n$ is $1$. Since each term after the first is $4$ times the preceding term, the value of $w$ is $9\left(4\right)$ when the value of $n$ is $2$. Therefore, the value of $w$ is $9\left(4\right)\left(4\right)$, or $9{\left(4\right)}^2$, when the value of $n$ is $3$. More generally, the value of $w$ is $9\left(4^{n-1}\right)$ for a given value of $n$. Therefore, the equation $w=9\left(4^{n-1}\right)$ gives $w$ in terms of $n$.

Choice A is incorrect. This equation describes a sequence for which the first term is $36$, rather than $9$, and each term after the first is $9$, rather than $4$, times the preceding term. 

Choice B is incorrect. This equation describes a sequence for which the first term is $4$, rather than $9$, and each term after the first is $9$, rather than $4$, times the preceding term.

Choice C is incorrect. This equation describes a sequence for which the first term is $36$, rather than $9$.

Choice D is correct. A y-intercept of a graph in the xy-plane is a point where the graph intersects the y-axis, or a point where $x=0$. Substituting $0$ for $x$ in the equation defining function $f$ yields $f\left(0\right)=a\left({2.2}^0+{2.2}^b\right)$, or $f\left(0\right)=a\left(1+{2.2}^b\right)$. So, the y-coordinate of the y-intercept of the graph is $a\left(1+{2.2}^b\right)$, or equivalently, $a+a{\left(2.2\right)}^b$. It's given that function $g$ is equivalent to function $f$, where $0<a<b$. It follows that $k={2.2}^b$. Since $a{\left(2.2\right)}^b$ can't be equal to $0$, the coefficient $a$ can't be equal to $a+a{\left(2.2\right)}^b$. Since $0<a$, the constant $k$, which is equal to ${2.2}^b$, can't be equal to $a+a{\left(2.2\right)}^b$. Therefore, function $g$ doesn't display the y-coordinate of the y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane as a constant or coefficient. It's also given that function $h$ is equivalent to function $f$, where $0<a<b$. The equation defining $f$ can be rewritten as $f\left(x\right)=a{\left(2.2\right)}^x+a{\left(2.2\right)}^b$. It follows that $m=a{\left(2.2\right)}^b$. Since $a{\left(2.2\right)}^b$ can't be equal to $0$, the coefficient $a$ can't be equal to $a+a{\left(2.2\right)}^b$. Since $0<a$, the constant $m$, which is equal to $a{\left(2.2\right)}^b$, can't be equal to $a+a{\left(2.2\right)}^b$. Therefore, function $h$ doesn't display the y-coordinate of the y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane as a constant or coefficient. Thus, neither function $g$ nor function $h$ displays the y-coordinate of the y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane as a constant or coefficient.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice B is correct. The vertex of the graph of a quadratic equation is where it reaches its minimum or maximum value. When a quadratic equation is written in the form $y=a{(x-h)}^2+k$, the vertex of the parabola represented by the equation is $(x,y)=(h,k)$. In the given equation $y=-16{(x-5.6)}^2+502$, the value of $h$ is $5.6$ and the value of $k$ is $502$. It follows that the vertex of the graph of this equation in the xy-plane is $(x,y)=(5.6,502)$. Additionally, since $a=-16$ in the given equation, the graph of the quadratic equation opens down, and the vertex represents a maximum. It’s given that the value of $y$ represents the estimated height, in feet, of the toy rocket $x$ seconds after it is launched into the air. Therefore, this toy rocket reaches an estimated maximum height of $502$ feet $5.6$ seconds after it is launched into the air.

Choice A is incorrect. The $16$ in the equation is an indicator of how narrow the graph of the equation is rather than where it reaches its maximum.

Choice C is incorrect. The $16$ in the equation is an indicator of how narrow the graph of the equation is rather than where it reaches its maximum.

Choice D is incorrect. This is an interpretation of the vertex of the graph of the equation $y=-16{(x-502)}^2+5.6$, not of the equation $y=-16{(x-5.6)}^2+502$.

Choice A is correct. The solution to the given system of equations can be found by solving the first equation for x, which gives , and substituting that value of x into the second equation which gives . Rewriting this equation by adding like terms and expanding gives . Subtracting from both sides of this equation gives . Adding to 2 to both sides of this equation gives . Therefore, it follows that . Substituting for y in the first equation yields . Adding to both sides of this equation yields . Therefore, the ordered pair is a solution to the given system of equations.

Choice B is incorrect. Substituting for x and for y in the first equation yields , or , which isn’t a true statement. Choice C is incorrect. Substituting for x and for y in the second equation yields , or , which isn’t a true statement. Choice D is incorrect. Substituting for x and for y in the second equation yields , or , which isn’t a true statement.